The Relationship Between Scientific Constants {© 01/02/20}

The following magical sequence proves that Rydberg's, Planck's, Newton's, Coulomb's, etc. constants and formulas are all related through energy, and also confirms that the neutronic ratio (Rₙ) is a fundamental part of all of them.

Note: All the input data in these calculations has been provided by CalQlata's Constants page.
All calculations are the sole copyright priority of Keith Dixon-Roche © 2018
Keith Dixon-Roche is also responsible for all the other web pages on this site related to atomic theory

Watch this, it's magic!

1) Questions:

Do you believe that Isaac Newton understood the mathematical laws of orbits?
Do you accept the Magnetic Constant?
μₒ = 4π . 1E-07 kg.m / C²
Do you accept the Permittivity Constant?
εₒ = 8.85418775855161E-12 C² / J.m
Permittivity is a ratio of electron magnetic charge displacement and electron magnetic field intensity, which is actually; C/m : J/C
εₒ = e² / 8.π.hꞌ = 8.85418775855161E-12 C² / J.m

2) You must therefore accept the coupling ratio

Magnetic [attraction] Force: Fₘ = G.m₁.m₂ / R² (Newton & Gilbert)
Electrical [repulsion] Force: Fₑ = k.q₁.q₂ / R² (Coulomb & Maxwell)
φ = Fₘ / Fₑ = G.mₚ.mₑ / k.e² = 4.40742111792334E-40

3) Newton's formula tells us that; g = v²/R = G.m/R²

From which, we get, when replacing 'v' with 'c':
R = G.mₚ / φ.c² = 2.81793795383896E-15 m (Rₙ!)
So, we know that anything orbiting a proton at 'c' must be orbiting at 2.81793795383896E-15 m
'Rₙ' is therefore a constant that represents the orbital radius of an electron travelling at 'c'.
Newton and Coulomb both said so!

4) The Magnetic Constant:

μₒ = 4π x 1E-07 kg.m/C²
but what is 1E-07?
mₑ.Rₙ/e² = 1.000000000000000E-07 kg.m/C²

Coincidence?

5) Let's just check!

Coulomb's constant:
k = μₒ.c² / 4π = 8.98755184732667E+09 kg.m³ / C².s²
Incorporating 4) above (μₒ) we find:
k = mₑ.c².Rₙ / 4π.e² = 8.98755184732667E+09 kg.m³ / C².s²

6) So, now we know the basis of Coulomb's constant:

k = (½.mₑ.c²) . (Rₙ/2π) / (e²) {kg.m³ / C².s² = kg.m²/s² . m / C² = J.m/C²}
Remember 'J.m/C²' when it comes to Planck's constant (h)

7) Let's check using Planck's constant:

h = √[π.mₑ.aₒ / εₒ] = 6.62607174469163E-34 kg.m²/s
Planck claimed its units as 'J.s'
However, a frequency ratio must be applied to his formula to achieve these units:
h = √[π.mₑ.aₒ / εₒ] . ƒ₁/ƒ₂ {kg.m²/s . s/s} (J.s)

8) If we expand εₒ in Planck's formula:

h = √[4.π².mₑ².c².aₒ.Rₙ] = 6.62607174469163E-34 kg.m²/s
which can be reorganised thus:
h = √[aₒ.(4.π)² . Rₙ] . ½.mₑ.c ⁽²⁾ #
= 6.62607174469163E-34 kg.m²/s
i.e. average radius x kinetic energy! [but there is something missing #].

9) So let's change Planck's formula a bit:

hꞌ = √[aₒ.(4.π)² . Rₙ] . ½.mₑ.c . v {kg.m²/s . m/s}
but what is v?
Let's try; vₒ = c . √[aₒ.(4.π)² . Rₙ] = 174090.866621084 m/s
hꞌ = 1.15353857232684E-28 kg.m³/s² {J.m}
but now the units work.

10) hꞌ also equals ...

hꞌ = ½.mₑ.c² . Rₙ {J.m}
validating the modified version of Planck's constant (remember Coulomb's constant J.m/C²)
In this case; 'Rₙ' is not only the orbital radius of the electron when travelling at 'c', it is also the coincident electro-magnetic amplitude
(which is the purpose of Planck's constant).

11) h vs hꞌ in the atom:

Now if we try both constants to calculate the kinetic energy of an orbiting electron; Eᴾ using h & E using hꞌ ...
It appears that Planck's constant (h) doesn't work but hꞌ does!
But we have established a definite link between Coulomb's and Planck's constants.

R (A) v KE Eᴾ (error) E (error)
6.2332E+08
3.1166E+08
2.0777E+08
1.5583E+08
1.2466E+08
1.0389E+08
8.9045E+07
7.7915E+07
6.9257E+07
6.2332E+07
5.6665E+07
5.1943E+07
2.8179E-15
5.6359E-15
8.4538E-15
1.1272E-14
1.4090E-14
1.6908E-14
1.9726E-14
2.2544E-14
2.5361E-14
2.8179E-14
3.0997E-14
3.3815E-14
299792459
211985280.7
173085256.9
149896229.5
134071263.5
122389758.9
113310898.8
105992640.4
99930819.7
94802699.6
90390827.4
86542628.5
4.0936E-14
2.0468E-14
1.3645E-14
1.0234E-14
8.1871E-15
6.8226E-15
5.8479E-15
5.1170E-15
4.5484E-15
4.0936E-15
3.7214E-15
3.4113E-15
1.12193E-11 (274.1)
3.96662E-12 (193.8)
2.15915E-12 (158.2)
1.40241E-12 (137.0)
1.00348E-12 (122.6)
7.63376E-13 (111.9)
6.05785E-13 (103.6)
4.95828E-13(96.90)
4.15529E-13 (91.36)
3.54785E-13 (86.67)
3.07522E-13 (82.64)
2.69894E-13 (79.12)
4.09356E-14 (1)
2.04678E-14 (1)
1.36452E-14 (1)
1.02339E-14 (1)
8.18711E-15 (1)
6.82259E-15 (1)
5.84794E-15 (1)
5.11694E-15 (1)
4.54840E-15 (1)
4.09356E-15 (1)
3.72141E-15 (1)
3.41130E-15 (1)
Planck’s Energy (Eᴾ = h.ƒ)
Note: the above (error) is a ratio and therefore a value of 1 represents zero error

12) Comparing Planck's minimum velocity (vₒ) with light-speed:

ξᵥ = vₒ:c
We can use this ratio (ξᵥ) to simplify Rydberg's wave number ...
R = mₑ.e⁴ / 8.εₒ².h³.c = 10973726.9561359 /m
which becomes:
R = 1 / aₒ.ξᵥ = 10973726.9561359 /m

13) Now if we apply Rydberg's wave number; R∞ ...

To his own energy constant:
Rᵧ = R.h.c.(Z.n)² = Rₙ/aₒ . ½.mₑ.c² = 2.17987197684936E-18 J
By an amazing coincidence, it just happens to be the kinetic energy of an electron orbiting at 'aₒ' metres {Ṯ = 33192.4K}.
So, Rydberg was telling us that he defined an electron's rest-state to occur when orbiting at 'aₒ'.

14) Rydberg (not Bohr) therefore defined 'aₒ':

aₒ = Rₙ . ½.mₑ.c² / Rᵧ {m}
So why do we refer to this dimension as Bohr's radius?
Because Bohr originally declared it to be the orbiting radius of a hydrogen electron at rest-mass; the only electron property he managed to calculate. He subsequently declared that electrons do not orbit (Quantum Theory) ...
... whereas Rydberg had actually told us that the orbital radius of an electron can be found from:
Rₑ / Rₙ = ½.mₑ.c² / KEₑ
which has subsequently proved to be correct, therefore Bohr was wrong, so we should be referring to 'aₒ' as the Rydberg radius.

15) Planck's mean [orbital] radius can be found thus:

Rₘ = Rₙ.ξᵥ = Rₙ / aₒ.R

hꞌ = ½.mₑ.c² . Rₙ {J.m}
and:
R = 1 / aₒ.ξᵥ
we have therefore, also established a definite working link between Planck's and Ryberg's constants.

16) Coulomb's constant is related to hꞌ thus:

hꞌ = e² / 8π.εₒ = ½.k.e² = 1.15353857232684E-28 {J.m}
It is interesting to note that using the modified version of Planck's constant we can find the fine-structure constant 'α':
2.hꞌ.εₒ = e² / 4π = 2.04272942122269E-39 C²

17) Electrical and magnetic forces must be equal when the electron is orbiting at 'c'

i.e. magnetic force = electrical force @ Rₙ
F = Rₙ².mₑ.c²/Rₙ³ = k.e²/Rₙ²
Rₙ = k.e² / mₑ.c² = 2.81793795383896E-15
{kg.m³.C² / C².s² / (kg.m²/s²) = m}
Confirming again that Coulomb said so.

18) and finally ...

kB = mₑ.c² / Y.Tₙ
i.e. the potential energy in a proton-electron pair at the time of its conversion to a neutron ...
... confirming that electrons orbit protons in circular paths.

To Conclude

After first discovering 'Rₙ' long-hand (iteratively), an unequivocal link between Newton's, Coulomb's, Planck's and Rydberg's constants has been established, all of which are dependent upon the neutronic radius.
'Rₙ' must, therefore be a real value that not only complies with known and accepted physical constants, it also validates those formula's in terms of their units and explains their meaning.
If we accept the validity of 'Rₙ' we must also accept the concept of circular orbits in atoms.
If we accept circular orbits in atoms, we must reject Quantum theory.

Moreover; Relativity invalidates the following accepted constants:
μₒ, εₒ, k, h, hꞌ, R, Rᵧ
because they all rely on 'Rₙ'
and, according to Relativity:
v = v / √[1+v/c] !
which means an electron could never achieve 'c'
(c = c/√2); so 'Rₙ' would be impossible
{c = c/√2 is of course nonsense because c ≠ c/√2}
Moreover, because 'Rₙ' proves it is impossible for an electron to travel at light-speed (c) in free-flight, photons cannot exist!